Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
9493140 | Journal of Algebra | 2005 | 37 Pages |
Abstract
Let A be a set of k⩾5 elements of an Abelian group G in which the order of the smallest nonzero subgroup is larger than 2kâ3. Then the number of different elements of G that can be written in the form a+aâ², where a,aâ²âA, aâ aâ², is at least 2kâ3, as it has been shown in [Gy. Károlyi, The ErdÅs-Heilbronn problem in Abelian groups, Israel J. Math. 139 (2004) 349-359]. Here we prove that the bound is attained if and only if the elements of A form an arithmetic progression in G, thus completing the solution of a problem of ErdÅs and Heilbronn. The proof is based on the so-called 'Combinatorial Nullstellensatz.'
Related Topics
Physical Sciences and Engineering
Mathematics
Algebra and Number Theory
Authors
Gyula Károlyi,